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G = Q32⋊C2order 64 = 26

2nd semidirect product of Q32 and C2 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C8.4D4, Q322C2, C4.15D8, C16.C22, SD322C2, C22.6D8, M5(2)⋊2C2, C8.11C23, D8.3C22, Q16.3C22, C4○D8.4C2, C4.12(C2×D4), C2.17(C2×D8), (C2×C4).48D4, (C2×Q16)⋊10C2, (C2×C8).24C22, 2-Sylow(ASigmaL(2,49)), SmallGroup(64,191)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — Q32⋊C2
C1C2C4C8C2×C8C2×Q16 — Q32⋊C2
C1C2C4C8 — Q32⋊C2
C1C2C2×C4C2×C8 — Q32⋊C2
C1C2C2C2C2C4C4C8 — Q32⋊C2

Generators and relations for Q32⋊C2
 G = < a,b,c | a16=c2=1, b2=a8, bab-1=a-1, cac=a9, bc=cb >

2C2
8C2
4C4
4C4
4C22
4C4
2D4
2Q8
2Q8
2Q8
4C2×C4
4Q8
4D4
4C2×C4
2SD16
2C2×Q8
2C4○D4
2Q16

Character table of Q32⋊C2

 class 12A2B2C4A4B4C4D4E8A8B8C16A16B16C16D
 size 1128228882244444
ρ11111111111111111    trivial
ρ211-1-11-111-111-1-11-11    linear of order 2
ρ3111-111-1-1-11111111    linear of order 2
ρ411-111-1-1-1111-1-11-11    linear of order 2
ρ511-1-11-11-1111-11-11-1    linear of order 2
ρ61111111-1-1111-1-1-1-1    linear of order 2
ρ711-111-1-11-111-11-11-1    linear of order 2
ρ8111-111-111111-1-1-1-1    linear of order 2
ρ922-202-2000-2-220000    orthogonal lifted from D4
ρ10222022000-2-2-20000    orthogonal lifted from D4
ρ1122-20-22000000-222-2    orthogonal lifted from D8
ρ122220-2-2000000-2-222    orthogonal lifted from D8
ρ1322-20-220000002-2-22    orthogonal lifted from D8
ρ142220-2-200000022-2-2    orthogonal lifted from D8
ρ154-40000000-222200000    symplectic faithful, Schur index 2
ρ164-4000000022-2200000    symplectic faithful, Schur index 2

Smallest permutation representation of Q32⋊C2
On 32 points
Generators in S32
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)
(1 26 9 18)(2 25 10 17)(3 24 11 32)(4 23 12 31)(5 22 13 30)(6 21 14 29)(7 20 15 28)(8 19 16 27)
(1 17)(2 26)(3 19)(4 28)(5 21)(6 30)(7 23)(8 32)(9 25)(10 18)(11 27)(12 20)(13 29)(14 22)(15 31)(16 24)

G:=sub<Sym(32)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32), (1,26,9,18)(2,25,10,17)(3,24,11,32)(4,23,12,31)(5,22,13,30)(6,21,14,29)(7,20,15,28)(8,19,16,27), (1,17)(2,26)(3,19)(4,28)(5,21)(6,30)(7,23)(8,32)(9,25)(10,18)(11,27)(12,20)(13,29)(14,22)(15,31)(16,24)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32), (1,26,9,18)(2,25,10,17)(3,24,11,32)(4,23,12,31)(5,22,13,30)(6,21,14,29)(7,20,15,28)(8,19,16,27), (1,17)(2,26)(3,19)(4,28)(5,21)(6,30)(7,23)(8,32)(9,25)(10,18)(11,27)(12,20)(13,29)(14,22)(15,31)(16,24) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)], [(1,26,9,18),(2,25,10,17),(3,24,11,32),(4,23,12,31),(5,22,13,30),(6,21,14,29),(7,20,15,28),(8,19,16,27)], [(1,17),(2,26),(3,19),(4,28),(5,21),(6,30),(7,23),(8,32),(9,25),(10,18),(11,27),(12,20),(13,29),(14,22),(15,31),(16,24)])

Q32⋊C2 is a maximal subgroup of
Q32⋊C4
 D8.D2p: D8.D4  D8.3D4  D8.12D4  C8.3D8  C8.5D8  SD32⋊S3  D8.9D6  SD32⋊D5 ...
 D2p.D8: D4○SD32  Q8○D16  Q32⋊S3  Q32⋊D5  Q32⋊D7 ...
 C8p.C23: D16⋊C22  C16.D6  C24.27C23  C16.D10  Q16.D10  C16.D14  Q16.D14 ...
Q32⋊C2 is a maximal quotient of
C23.39D8  C23.41D8  M5(2)⋊1C4  SD323C4  Q324C4  Q167D4  C162D4  D81Q8  Q16⋊Q8  Q16.Q8  C22.D16  C23.50D8  C23.20D8  C8.12SD16  C8.14SD16  C16⋊Q8
 C16.D2p: C16.D4  C8.7D8  C16.D6  SD32⋊S3  Q32⋊S3  C16.D10  SD32⋊D5  Q32⋊D5 ...
 D8.D2p: D8.10D4  D8.4D4  D8.9D6  C40.31C23  C56.31C23 ...
 Q16.D2p: Q16.8D4  Q16.4D4  Q16.5D4  C24.27C23  Q16.D10  Q16.D14 ...

Matrix representation of Q32⋊C2 in GL4(𝔽7) generated by

2354
0321
2533
4446
,
3532
1442
1636
5524
,
6014
0545
0455
0665
G:=sub<GL(4,GF(7))| [2,0,2,4,3,3,5,4,5,2,3,4,4,1,3,6],[3,1,1,5,5,4,6,5,3,4,3,2,2,2,6,4],[6,0,0,0,0,5,4,6,1,4,5,6,4,5,5,5] >;

Q32⋊C2 in GAP, Magma, Sage, TeX

Q_{32}\rtimes C_2
% in TeX

G:=Group("Q32:C2");
// GroupNames label

G:=SmallGroup(64,191);
// by ID

G=gap.SmallGroup(64,191);
# by ID

G:=PCGroup([6,-2,2,2,-2,-2,-2,121,199,650,579,297,165,1444,730,88]);
// Polycyclic

G:=Group<a,b,c|a^16=c^2=1,b^2=a^8,b*a*b^-1=a^-1,c*a*c=a^9,b*c=c*b>;
// generators/relations

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Subgroup lattice of Q32⋊C2 in TeX
Character table of Q32⋊C2 in TeX

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